A critical issue for users of Markov Chain Monte Carlo (MCMC) methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest. Research into methods of computing theoretical convergence bounds holds promise for the future but currently has yielded relatively little that is of practical use in applied work. Consequently, most MCMC users address the convergence problem by applying diagnostic tools to the output produced by running their samplers. After giving a brief overview of the area, we provide an expository review of thirteen convergence diagnostics, describing the theoretical basis and practical implementation of each. We then compare their performance in two simple models and conclude that all the methods can fail to detect the sorts of convergence failure they were designed to identify. We thus recommend a combination of strategies aimed at evaluating and accelerating MCMC sampler conver...

In this paper we derive conditions for geometric ergodicity of the random walk-based Metropolis algorithm on R k . We show that at least exponentially light tails of the target density is a necessity. This extends the one-dimensional result of (Mengersen and Tweedie, 1996). For sub-exponential target densities we characterize the geometrically ergodic algorithms and we derive a practical sufficient condition which is stable under addition and multiplication. This condition is especially satisfied for the class of densities considered in (Roberts and Tweedie, 1996).

this paper, we analyse theoretical properties of the slice sampler. We find that the algorithm has extremely robust geometric ergodicity properties. For the case of just one auxiliary variable, we demonstrate that the algorithm is stochastic monotone, and deduce analytic bounds on the total variation distance from stationarity of the method using Foster-Lyapunov drift condition methodology. 1. Introduction.

This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain methods. Each method is discussed giving an outline of the basic supporting theory and particular features of the technique. Conclusions are drawn concerning the relative merits of the methods based on the discussion and their application to three examples. The following broad recommendations are made. Asymptotic methods should only be considered in contexts where the integrand has a dominant peak with approximate ellipsoidal symmetry. Importance sampling, and preferably adaptive importance sampling, based on a multivariate Student should be used instead of asymptotics methods in such a context. Multiple quadrature, and in particular subregion adaptive integration, are the algorithms of choice for...

This chapter develops Markov Chain Monte Carlo (MCMC) methods for Bayesian inference in continuous-time asset pricing models. The Bayesian solution to the inference problem is the distribution of parameters and latent variables conditional on observed data, and MCMC methods provide a tool for exploring these high-dimensional, complex distributions. We first provide a description of the foundations and mechanics of MCMC algorithms. This includes a discussion of the Clifford-Hammersley theorem, the Gibbs sampler, the Metropolis-Hastings algorithm, and theoretical convergence properties of MCMC algorithms. We next provide a tutorial on building MCMC algorithms for a range of continuous-time asset pricing models. We include detailed examples for equity price models, option pricing models, term structure models, and regime-switching models. Finally, we discuss the issue of sequential Bayesian inference, both for parameters and state variables.

This chapter discusses Markov Chain Monte Carlo (MCMC) based methods for es- timating continuous-time asset pricing models. We describe the Bayesian approach to empirical asset pricing, the mechanics of MCMC algorithms and the strong theoretical underpinnings of MCMC algorithms. We provide a tutorial on building MCMC algo- rithms and show how to estimate equity price models with factors such as stochastic expected returns, stochastic volatility and jumps, multi-factor term structure models with stochastic volatility, time-varying central tenclancy or jumps and regime switching models.

In Bayesian inference, a Bayes factor is defined as the ratio of posterior odds versus prior odds where posterior odds is simply a ratio of the normalizing constants of two posterior densities. In many practical problems, the two posteriors have different dimensions. For such cases, the current Monte Carlo methods such as the bridge sampling method (Meng and Wong 1996), the path sampling method (Gelman and Meng 1994), and the ratio importance sampling method (Chen and Shao 1994) cannot directly be applied. In this article, we extend importance sampling, bridge sampling, and ratio importance sampling to problems of different dimensions. Then we find global optimal importance sampling, bridge sampling, and ratio importance sampling in the sense of minimizing asymptotic relative mean-square errors of estimators. Implementation algorithms, which can asymptotically achieve the optimal simulation errors, are developed and two illustrative examples are also provided.

8> R f(`)d`. To marginalize, say for ` i ; requires h(` i ) = R f(`)d` (i) = R f(`)d` where ` (i) denotes all components of ` save ` i : To obtain Eg(` i ) requires similar integration; to obtain the marginal distribution of say g(`) or its expectation requires similar integration. When p is large (as it will be in the applications we envision) such integration is analytically infeasible (the so-called curse of dimensionality*). Gibbs sampling provides a Monte Carlo approach for carrying out such integrations. In what sorts of settings would we have need to mar

this paper we develop schemes along both lines to apply to the analysis of GARCH (generalized autoregressive conditional heteroskedasticity) models for daily exchange rate data. KEY WORDS: Exchange rate data; General dynamic model; Markov chain Monte Carlo; Metropolis; State space model. 1 INTRODUCTION ARCH (autoregressive conditional heteroskedasticity) models have achieved a considerable following in the econometrics and finance literature since their introduction by Engle (1982). An ARCH model is a discrete stochastic process with the characteristic feature that the variance at time t is some time-varying function of the time t \Gamma 1 information set. Below, in (1) we give a more specific formal definition. Applications and studies were stimulated by the generalisation to GARCH models in Bollerslev (1986). There are now over 300 papers in the mainstream statistics and econometrics journals discussing theoretical properties of many formulations of GARCH model as well as numerous applications. An excellent survey of the literature is Bollerslev et al. (1992); some of the review material later in this section is based on that article. The contribution of this paper is twofold. Firstly, novel combinations of Markov chain Monte Carlo techniques are developed. Though prompted by, and developed here specifically for, dynamic GARCH models, the algorithms are useful for a wide range of non-standard sequential analyses, dynamic or not. The methods build on and are related to Markov chain Monte Carlo techniques discussed in Carlin, Polson and Stoffer (1992), Jacquier, Polson and Rossi (1994 and 1995), and Tierney (1994). Secondly, the class of (multivariate) GARCH models is extended to a dynamic setting thereby allowing much greater modelling flexibility. We note that dynam...

This paper shows how coupling methodology can be used to give precise, a priori bounds on the convergence time of Markov chain Monte Carlo algorithms for which a partial order exists on the state space which is preserved by the Markov chain transitions. This methodology is applied to give a bound on the convergence time of the random scan Gibbs sampler used in the Bayesian restoration of an image of N pixels. For our algorithm, in which only one pixel is updated at each iteration, the bound is a constant times N². The